The seminar meets regularly on Wednesdays at 4pm in Eckhart 202. We also have special seminars during other days. To subscribe or unsubscribe the email list, you may either go to Camp/PDE email list or contact Cornelia Mihaila.
Scalar conservation law equations develop jump discontinuities even when the initial data is smooth. Ideally, we would expect these discontinuities to be confined to a collection of codimension-one surfaces, and the solution to be relatively smoother away from these jumps. The picture is less clear for rough initial data which is merely bounded. While a linear transport equation may have arbitrarily rough solutions, genuinely nonlinear conservation laws have a subtle regularization effect. We prove that the entropy solution will become immediately continuous outside of a codimension-one rectifiable set, that all entropy dissipation is concentrated on the closure of this set, and that the L∞ norm of the solution decays at a certain rate as t goes to infinity.
Mathematical models supporting diffusion adjustments in subregions of the domain are relevant in several fields of research. This is a typical issue, for instance, in imaging enhancements and by now a comprehensive variational theory for treating functionals satisfying p-growth conditions has been established. This encompasses, in particular, problems ruled by the so called p(x)-laplacian. Until quite recently, The corresponding non-variational theory was completely open, despite of its potential applicabilities, and in this talk I will describe some lately efforts towards launching such a theory. We will introduce a new class of non-divergence form elliptic operators whose degree of degeneracy/singularity varies accordantly to a prescribed power law. Under rather general conditions, we prove viscosity solutions to variable exponent fully nonlinear elliptic equations are differentiable, with appropriate (universal) estimates. This result opens a number of new lines of investigation and I'll describe some of these.
We study a model for combustion on a boundary by studying solutions of a fractional unstable obstacle problem. We analyze the free boundary and prove an upper bound for the Hausdorff dimension of the singular set. We also show that certain symmetric solutions are either stable or unstable depending on the parameter s, of the fractional Laplacian. This is joint work with Mark Allen.
The talk is concerned with a partial regularity in time result for solutions of the space-homogeneous Landau-Coulomb equation. More precisely, we will see that the Hausdorff dimension of the set of times at which a solution is not locally (in time) bounded (in time and the velocity variable) is at most 1/2+0. Our proof relies on energy estimates satisfied by relative entropies of solutions. These estimates are derived by revisiting the entropy dissipation estimate of Desvillettes (2015). They allow one to apply De Giorgi’s iterative truncation procedure. This is a joint work with F. Golse, M. Gualdani and A. F. Vasseur.
We study nonlocal operators that generate anisotropic jump processes, such as a jump process that behaves like a stable process in each direction but with a different index of stability. Its generator is the sum of one-dimensional fractional Laplace operators with different orders of differentiability. We study such operators in the general framework of bounded measurable coefficients. The objective of this talk is to discuss regularity results for weak solutions to corresponding integro-differential equations. Joint work with Moritz Kassmann.
Abstract: The original mass transport problem, formulated by Gaspard Monge in 1781, asks to find the optimal volume preserving map between two given sets of equal volume, where optimality is measured against a cost functional given by the integral of a cost density. After reviewing some aspects of this classical problem, I will describe joint work with Nick Barron and Bob Jensen (Loyola University Chicago) leading to a duality theory for the case of relaxed L∞ cost functionals acting on probability measures with prescribed marginals. Several formulations of the dual problem are obtained using quasiconvex analysis and PDE techniques.